I understand that there are different definitions of finite. They are all equivalent over ZFC, and some of them are even equivalent over ZF, but not over weaker theories than ZF(C). I would like some texts that define various notions of finite, and develop the theory quite a bit.
2026-04-02 04:44:06.1775105046
Reference request for different definitions of finite
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I don't know of a book that treats this specifically - although there should be in my opinion - but below is a list of relevant papers, in chronological order. Personally I recommend starting with the introductions of the more recent papers.
I've made this CW, and I encourage others to add to it (but please keep the chronological ordering, I think that lends some clarity):
1958: Levy, The independence of various definitions of finiteness.
1988: Spišiak/Vojtáš, Dependences between definitions of finiteness.
2004: Cruz/Dzhafarov/Hall, Definitions of finiteness based on order properties.
2019: Brot/Cao/Fernandez-Breton, Finiteness classes arising from Ramsey-theoretic statements in set theory without choice.